Noncompact Worldline Topologies

• Noncompact worldline topologies are geometrical representations of particle trajectories on unbounded domains, enabling seamless treatment of asymptotic states and LSZ reduction.
• They simplify complex computations by using a master integral that captures contributions from various Feynman diagram topologies without segmenting integration limits.
• Their application extends to quantum gravity and noncommutative geometry, providing a unified framework that automates boundary conditions and enhances gauge invariance.

Noncompact worldline topologies refer to worldline geometries—parametrizations of particle trajectories or one-dimensional manifolds—whose domains are not confined to compact intervals or circles. This concept arises in various theoretical constructions, notably in worldline path integrals, quantum field theory (QFT), quantum gravity, geometric analysis, and noncommutative geometry. The central significance of noncompact worldline topologies is their role in automating boundary conditions, encoding asymptotic states, unifying summation over topologies in perturbation theory, and underpinning key mathematical invariants and algebraic structures.

1. Definitions and Formalism

The worldline formalism treats physical processes as integrals over particle trajectories (worldlines) rather than positions or fields. The topology of these worldlines—compact (closed loop/circle), interval (finite segment), half-line, or full line—directly affects the analytic and physical structure of amplitudes and propagators.

• Compact worldline: Typically parameterized on a circle $S^1$. Common in closed-loop (e.g., vacuum polarization) computations.
• Noncompact worldline: Extended interval, half-line $[0, \infty)$, or full line $\mathbb{R}$. Used for propagators (open lines), LSZ reductions, and group averaging schemes.

Fundamental path integral expressions:

• Compact (one-loop):

$$\Gamma[\{k_i, \varepsilon_i\}] = (-ie)^N \int_0^\infty \frac{dT}{T} (4\pi T)^{-D/2} e^{-m^2T} \prod_{i=1}^{N} \int_0^T d\tau_i \, \exp\left\{\Sigma_{i,j=1}^{N} [ \ldots ] \right\}$$

• Noncompact (propagator):

$$D^{xx'}[A] = \int_0^\infty dT \, e^{-m^2T} \int_{x(0)=x'}^{x(T)=x} \mathcal{D}x(\tau) \, \exp \left\{ -\int_0^T d\tau [\frac{1}{2} \dot{x}^2 + ie \dot{x} \cdot A(x(\tau)) ] \right\}$$

Automating outgoing/incoming (on-shell) states is achieved by sending endpoints to infinity in the noncompact topology, inherently implementing LSZ reduction (Kim, 7 Sep 2025).

2. Computational Advantages and Integration Techniques

Worldline approaches utilizing noncompact topologies allow a single "master" integral to represent contributions from many Feynman diagrams differing in topology (ordering and connectivity of vertices). For photon amplitudes, the noncompact integration domain covers all possible insertion points seamlessly; no sectoring into ordered subspaces is required.

• The absence of ordering means that integrals involving signum and absolute value functions (from Green's functions) must be handled globally, leading to novel integration challenges.
• Integration-by-parts and operator techniques employ representations of inverse derivatives in terms of Bernoulli polynomials:

$$\langle u_1 | \partial_P^{-n} | u_2 \rangle = -\frac{1}{n!} B_n(|u_1 - u_2|) \mathrm{sgn}^n(u_1-u_2)$$

(Edwards et al., 2021) Integrals over continuous, noncompact moduli spaces yield connections to number theory, including Bernoulli numbers and multiple zeta values.

Recursive integration identities and expansions in inverse derivatives facilitate the evaluation of worldline integrals without decomposing into Feynman sub-diagrams. These techniques are crucial for extracting closed-form results for multi-photon and multi-loop amplitudes (Ahmadiniaz et al., 2022, Ahmadiniaz et al., 10 Jul 2024).

3. Physical Interpretation and Boundary Conditions

Topological choices for the worldline domain determine how asymptotic states are encoded:

• Interval ([0, T]): Standard off-shell propagator; LSZ reduction must be applied post-integral to obtain on-shell S-matrix elements.
• Half-line: One boundary at infinity, fixing the external momentum; the path integral projects directly onto on-shell states for one leg.
• Full line ($\mathbb{R}$): Both endpoints at infinity, representing full on-shell amplitudes, with external legs automatically amputated.

This automates the treatment of asymptotic states and reduces the need for manual manipulations such as additional Fourier transforms and derivatives. In phase space worldline formalism, noncompact topology simplifies Feynman rules and propagator structure, leading to cubic vertex interactions and manifest gauge invariance (Kim, 7 Sep 2025).

The boundary conditions implemented via noncompact topologies are also central in quantum gravity path integrals—where integration over the proper time parameter $T$ on $\mathbb{R}$ enforces Hamiltonian constraints via group averaging, yielding abelian operator algebras suitable for baby universe constructions (Casali et al., 2021).

4. Noncommutative Geometry and Fuzzy Topologies

Noncompact worldline topologies are extended to quantum deformations and noncommutative geometry. The $\kappa$-Poincaré algebra induces noncommutativity, not just on spacetime points but on the space of worldlines themselves. Coordinates parametrizing each worldline (e.g., $(\eta^i, y^i)$ relating to velocity and impact positions) obey nontrivial Heisenberg–Weyl relations:

$[\hat{q}^a, \hat{p}^b] = (i/\kappa) \delta_{ab}$

This renders worldlines "fuzzy," so intersection events (spacetime points) emerge as statistical phenomena with nonzero variance in impact parameters (Ballesteros et al., 2021). Thus, spacetime can be reconstructed from the quantum geometry of the ensemble of worldlines, with fundamental events acquiring probabilistic widths.

5. Worldline Topology Transitions in Gravity and Geometric Analysis

Noncompact worldline topology also appears in spacetime engineering. In Einstein–Maxwell–Kalb–Ramond gravity coupled to lightlike branes (LL-branes), noncompact segments (throats) connect regions of compact (Bertotti–Robinson) and decompactified (Reissner–Nordström–de Sitter) spacetime. The dynamics of LL-branes induces compactification/decompactification transitions, altering the topological character of worldline segments and the global geometry (Guendelman et al., 2010).

From the standpoint of geometric analysis, noncompact manifolds require special attention to global invariants such as the Yamabe constant. The continuity of the Yamabe constant (and its asymptotic value) with respect to fine $C^2$-topology ensures stable classification of worldline topologies, especially as they relate to the "ends" or infinity of the manifold (Große et al., 2012).

6. Applications in Perturbation Theory and Gauge Theories

In gauge theory (including noncommutative $U(1)$), noncompact worldline topologies facilitate:

• Summation over planar and nonplanar diagrams via phase space integrals (Ahmadiniaz et al., 2015).
• Universal separation of divergences (UV in planar, IR in nonplanar sectors).
• Homogeneous, string-inspired Feynman rules; the independence of effective action calculations from choices of worldline Green's function (Dirichlet or periodic boundary conditions).

In multiloop and non-abelian contexts (e.g., Yang–Mills, gravity), worldline methods employing noncompact topologies provide a uniform, cubicized computational framework across theories (Kim, 7 Sep 2025).

7. Mathematical Structures and Future Research Directions

The paper of noncompact worldline topologies continuously generates new mathematical techniques and challenges:

• Operator identities on the circle, connections to Bernoulli polynomials, frameworks for handling moduli integration.
• Recursive construction of exactly solvable worldline instanton models via deformation functions, yielding a spectrum of topologies with analytic control over noncompact configurations (Akal, 2018).
• Open questions concerning the classification and completeness of invariants at infinity, gaps in the spectrum of Yamabe constants, and the full algebraic characterization of superselected sectors in quantum gravity.

Future work aims to extend integration schemes to field-dependent Green’s functions, combined backgrounds (e.g., plane-wave plus constant field), and higher-loop structures, and to clarify the emergence and classification of fuzzy spacetime events.

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Noncompact worldline topologies fundamentally serve as both computational and conceptual bridges in high-energy physics, geometric analysis, and quantum gravity. Their paper unifies the treatment of asymptotic states, automates boundary conditions, and underpins deeper algebraic and geometric structures central to perturbative and nonperturbative approaches.